Ab initio molecular dynamics: Theory and Implementation

“unbound electrons” dissolved in liquid condensed matter (localizing as F–centers,polarons, bipolarons, etc.) — the time step difference of about a factor of ten confirmsthe crude estimate given above. In a Born–Oppenheimer simulation 569 ofagain K x·(KCl) 1−x but up to a higher concentration of unbound electrons the timestep used was 0.5 fs.The time–scale advantage of Born–Oppenheimer vs. Car–Parrinello dynamicsbecomes more evident if the nuclear dynamics becomes fairly slow, such as in liquidsodium 343 or selenium 331 where a time step of 3 fs was used. This establishesthe above–mentioned order of magnitude advantage of Born–Oppenheimer vs. Car–Parrinello dynamics in advantageous cases. However, it has to be taken into accountthat in simulations 331 with such a large time step dynamical information is limitedto about 10 THz, which corresponds to frequencies below roughly 500 cm −1 . Inorder to resolve vibrations in molecular systems with stiff covalent bonds the timestep has to be decreased to less than a femtosecond (see the estimate given above)also in Born–Oppenheimer dynamics.The comparison of the overall performance of Car–Parrinello and Born–Oppenheimer molecular dynamics in terms of computer time is a delicate issue.For instance it depends crucially on the choice made concerning the accuracy ofthe conservation of the energy E cons as defined in Eq. (48). Thus, this issue is tosome extend subject of “personal taste” as to what is considered to be a “sufficientlyaccurate” energy conservation. In addition, this comparison might todifferent conclusions as a function of system size. In order to nevertheless shedlight on this point, microcanonical simulations of 8 silicon atoms were performedwith various parameters using Car–Parrinello and Born–Oppenheimer moleculardynamics as implemented in the CPMD package 142 . This large–gap system wasinitially extremely well equilibrated and the runs were extended to 8 ps (and afew to 12 ps with no noticeable difference) at a temperature of about 360–370 K(with ±80 K root–mean–square fluctuations). The wavefunction was expanded upto E cut = 10 Ry at the Γ–point of a simple cubic supercell and LDA was usedto describe the interactions. In both cases the velocity Verlet scheme was used tointegrate the equations of motion, see Eqs. (231). It is noted in passing that alsothe velocity Verlet algorithm 638 allows for stable integration of the equations ofmotion contrary to the statements in Ref. 513 (see Sect. 3.4 and Figs. 4–5).In Car–Parrinello molecular dynamics two different time steps were used, 5 a.u.and 10 a.u. (corresponding to about 0.24 fs), in conjunction with a fictitious electronmass of µ = 400 a.u.; this mass parameter is certainly not optimized and thusthe time step could be increased furthermore. Also the largest time step lead toperfect adiabaticity (similar to the one documented in Fig. 3), i.e. E phys Eq. (49)and T e Eq. (51) did not show a systematic drift relative to the energy scale setby the variations of V e Eq. (50). Within Born–Oppenheimer molecular dynamicsthe minimization of the energy functional was done using the highly efficient DIIS(direct inversion in the iterative subspace) scheme using 10 “history vectors”, seeSect. 3.6. In this case, the time step was either 10 a.u. or 100 a.u. and threeconvergence criteria were used; note that the large time step corresponding to2.4 fs is already at the limit to be used to investigate typical molecular systems(with frequencies up to 3–4000 cm −1 ). The convergence criterion is based on the29